A bicycle odometer (which measures disu 3th yckl z6sdt:g5 2)ua4vz+tance traveled) is attached near the wheel hub and is designed f:kt2+hgl vzdt5)cu 3zyua 46sor 27-inch wheels. What happens if you use it on a bicycle with 24-inch wheels?

Suppose a disk rotates at constant angular vt xq -56ty)kyc:( ebqhelocity. Does a point on the rim have radial and/or tangential acceleration? If the disk’s angular velocity increases uniformly, does the point have radial and/qt: 6te )cqx-y5b k(hyor tangential acceleration? For which cases would the magnitude of either component of linear acceleration change?

Could a nonrigid body be described by a single value of the anrz/z dpg*q9 d)e)s*ijgular velocity $\omega$ Explain.

Can a small force ever exert a greater torque than a larger foh3u+q;*eho igh lg)05v e)mms rce? Explain.

If a force $\vec{F}$ acts on an object such that its lever arm is zero, does it have any effect on the object’s motion? Explain.

Why is it more difficult to do a sit-up with your hands behind yoc+ k;jjuc,d5)mxkmx qsg3n8r6sz.9bvt:qf 0 ur head than when your arms are stretched out in fronruc .+ms ;mj3kn5)vq,b:xkg 0jq9czxdt 8f 6st of you? A diagram may help you to answer this.

A 21-speed bicycle has seven sprockets at the rear wheel and three at the v*i37pz u o9za vlhf6(pedal cranks. In which gear is it harder to pedal, a small rearzv*7 va (up6flz9o3ih sprocket or a large rear sprocket? Why? In which gear is it harder to pedal, a small front sprocket or a large front sprocket? Why?

Mammals that depend on being able to run fast have slender loa m c,qf vr7.ie4vs8oo z)j 9:2sqsqc8wer legs with flesh and muscle concentrated high, close to the body (Fig. 8–34). On the basis of rotational dynamics, explain why this distribution of v4fcoqsc7mv)8 qz j ,qe:.as8 ir92somass is advantageous.

Why do tightrope walkers (Fig. 8–3l8l7zs oalx,s*w0hr2 5) carry a long, narrow beam?

If the net force on a syst nso -*aby0j agg2piwe*5 65ismt+b1f em is zero, is the net torque also zero? If the net torque on a**iej2m+sw0 yoa5 i5an1 bfbspg-g t6 system is zero, is the net force zero?

Two inclines have the same height but make different angles with tctj;a ,ud.)+0tdyo0x zkw :mr8 ofil 5he horizontal. The same steel ball is rolled down each ifi5)oz ;uwl ,axd+d:t0.t8y 0krjo mc ncline. On which incline will the speed of the ball at the bottom be greater? Explain.

Two solid spheres simultaneously start rolling (from rest) down an incline. .si,q uwoyu 2,One sphere has twice the radius and twice the mass of the other. Which reaches the bottom of the incline firstsui2,w., y quo? Which has the greater speed there? Which has the greater total kinetic energy at the bottom?

A sphere and a cylinder have the same radius and the1t6ox6rkoj*i same mass. They start from rest at the top of an 1ok t6rj*oi 6xincline. Which reaches the bottom first? Which has the greater speed at the bottom? Which has the greater total kinetic energy at the bottom? Which has the greater rotational KE?

We claim that momentum ag3w ork)i9eeh uqn+6 .0dct* jnd angular momentum are conserved. Yet most moving or rotating objects eve)grdwue6 n0ck h.*i+qj 3 9eotntually slow down and stop. Explain.

If there were a great migration of people toward the Earth’s equator, how would fl lzfec34o-/ v9.zvhor( 6 o( j)cxurthis affect the length of thofou6z39 vllefrr(x 4 /c)ov-(zj c.he day?

Can the diver of Fig. 8–29 do a somersault without t.* y ,:c9i6hdi1fbfm:gwy zn having any initial rotation when she leaves the boariw cbdf yi.mg*,1:ty6h znf9:d?

The moment of inertia of a rotating solid disk about an axis through its br8u c vel;bb(t,;sz0as87mccenter of mass issa, mbb;b0v7c (c zsur;8lte8 $\frac{1}{2}WR^2$ (Fig. 8–21c). Suppose instead that the axis of rotation passes through a point on the edge of the disk. Will the moment of inertia be the same, larger, or smaller?

Suppose you are sitting on a rotating stool holding a 2-kg mass in each outst,-t;i4bi ol4( gqr a*en8l6xrnf7ld oretched hand. If you suddenly drop the masses, will your angular velocity increase, decrease, or( tni-xfgaq64 4rrn7o ed 8*;,olilbl stay the same? Explain.

Two spheres look identical and have the same mass. However, one is hollow a51ki*fniu/ r/lvex qm2tzj 9* nd the other is solid. D91v/2*t u l r*5nxj qeiiz/mfkescribe an experiment to determine which is which.

The angular velocity of a wheel rotating on a horizonta- v pk e)o;27 jm66in(. nuubwnma:jbul axle points west. In what direction is the linear velocityub: nm7.v wa ;- (nmno6j6j)ibp2euku of a point on the top of the wheel? If the angular acceleration points east, describe the tangential linear acceleration of this point at the top of the wheel. Is the angular speed increasing or decreasing?

Suppose you are standing on the edge of a large freely cwlutq jz8 tx,f 3+i 4u7.q(hsrotating turntable. What happens if you waltutqqj7w+u,8f(s .iz4cx3lh k toward the center?

A shortstop may leap into the air to catch a ball and throw it quickly. tqp zk *y6x-g41k1am sn)ts 6yAs he throws the ball, the upper part of his body rotates. If you look quickly you -1sxs*gp4 kky6 t1)n atm6yzqwill notice that his hips and legs rotate in the opposite direction (Fig. 8–36). Explain.

On the basis of the law of conservation of+7irp z-ksx: a angular momentum, discuss why a helicopter must have more than one rotor (or propeller). Discuss one or more7apr :s kxiz+- ways the second propeller can operate to keep the helicopter stable.

  Express the following angles in radian 2 a0fwl.vvhx/al(y1ns: (a) 30 $^{\circ} $, (b) 57 $^{\circ} $, (c) 90 $^{\circ} $, (d) 360 $^{\circ} $, and (e) 420 $^{\circ} $. Give as numerical values and as fractions of $\pi$.(Round to two decimal places)
(a)   $rad$ (b)   $rad$ (c)    $rad$ (d)    $rad$ (e)    $rad$

  Eclipses happen on Earth because of an amazing coinmt3d36s1xjl o k: g/qacidence. Calculate, using the information inside the Front Cover, the angular diameters (in radians) ofom/x1 d:jgs3l tqa3 k6 the Sun and the Moon, as seen on Earth.
Sun =    $rad$ Moon =    $rad$

  A laser beam is directed at the Moon, 380,000 kmjmiv ff,(o; op7hlwd* h-,yf; from Earth. The beam diverges at an angl pfd,; jf*(hl y;o h-7viw,mofe $\theta$ (Fig. 8–37) of $1.4\times10^{-5}$ rad What diameter spot will it make on the Moon?    m

  The blades in a blend db.1t-g q(yhber rotate at a rate of 6500 rpm. When the motor is turned off during operation, the blades sl1hb-q gdbt(y. ow to rest in 3.0 s. What is the angular acceleration as the blades slow down?    $rad/s^2$

  A child rolls a ball on a level floor 3.5 m to another childa tt-nr:joe, :pxu5j9. If the ball makes 15.0 revolutions, what is it:puea-rtj,o x9: tjn 5s diameter?    m

  A bicycle with tires 6-x /g xfp *eao(.)bw:2febmyu8 cm in diameter travels 8.0 km. How many revolutions do the wheels ma)2xabeb e* o:gmy p-uw/f.(fxke?    $rev$

  (a) A grinding wheel 0.35 m in diameter rotates at 2500 rpm. Calcvha6)l 7k7v hjulate its angular ha6)7vk jh 7vlvelocity in $rad/s$ $\omega$ =    $rad/sec$
(b) What are the linear speed and acceleration of a point on the edge of the grinding wheel? v =    $m/s$ $a_R$ =    $ m/s^2$

  A rotating merry-go-round makes one complete revolution in 4.0 s (Fig. ari5tr8n2hwsc0ig02jr.z + d8–38). (a) What is r+28nardsi .0 jhwtiz 50 2crgthe linear speed of a child seated 1.2 m from the center?    $m/s$
(b) What is her acceleration (give components)?    $m/s^2$  ----待选择----towardsaways  the center

  Calculate the angular velocity of the Earth 9p2i gsn+.7iguch mmj7t .-eu(a) in its orbit around the Sun    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear speed-xz eu,5m sc)o of a point
(a) on the equator,    $m/s$
(b) on the Arctic Circle (latitude 66.5$^{\circ} $ N),    $m/s$
(c) at a latitude of 45.0$^{\circ} $ N, due to the Earth’s rotation?    $m/s$

  How fast (in rpm) must ax8zowzh ailunj 8 838u. r-ao2 centrifuge rotate if a particle 7.0 cm from the axis of rotation is to experience 38a8j-u i uxh.nlz 8z o2ar8owan acceleration of 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel accelerates uniformly about its center from 130 rpm to 2 /jyv2pzh9zz:80 rpm in 4.0 s. Det2 pzh9zz:v jy/ermine
(a) its angular acceleration,$\approx$    $rad/s^2$(Round to one decimal places)
(b) the radial and tangential components of the linear acceleration of a point on the edge of the wheel 2.0 s after it has started accelerating. $a_R$    $m/s^2$ $a_{tan}$    $m/s^2$

A turntable of radiuk3u:h;zk ic6, 34 rjbdyx j;vrs $R_1$ is turned by a circular rubber roller of radius $R_2$ in contact with it at their outer edges. What is the ratio of their angular velocities, $\omega_1$ / $\omega_2$

  In traveling to the Moon,d0wa mo al2li rz,vb-s),u- tu-g57px astronauts aboard the Apollo spacecraft put themselves into a slow rotation to distribute the Sun’s energy evenly. At the start of their trip, they accegua zpxi7-rlt5)2-o,,-s v ulbmw0da lerated from no rotation to 1.0 revolution every minute during a 12-min time interval. The spacecraft can be thought of as a cylinder with a diameter of 8.5 m. Determine
(a) the angular acceleration, $\approx$    $rad/s^2$
(b) the radial and tangential components of the linear acceleration of a point on the skin of the ship 5.0 min after it started this acceleration. $a_{tan}$ =    $ \times10^{ -4}$ $m/s^2$ $a_{rad}$ =    $ \times10^{ -3}$ $m/s^2$

  A centrifuge accelerates uniformly from rest to 15,000 rpm ind b)phcave4.n34f 7 uru+p)xy 220 s. Through how many revolutions did it turn in this timefu)hen3r xvy7+p u) c4b4 dpa.?    $rev$

  An automobile engine slows down from 4500 rpm to 1200 rpm ino9+yy:w jl, tz wkmqwj4: zu6) 2.5 s. Calculate
(a) its angular acceleration, assumed constant,    $rad/s^2$
(b) the total number of revolutions the engine makes in this time.    $rev$

  Pilots can be tested for the stresses of flying highspeed jets in a whirling “h,mh/ jyo5 3wuijufd 35uman centrifuge,” wh5jd/uj,wfy3 h53oi u mich takes 1.0 min to turn through 20 complete revolutions before reaching its final speed.
(a) What was its angular acceleration (assumed constant),    $rev/min^2$
(b) what was its final angular speed in rpm?    $rpm$

  A wheel 33 cm in diameter accelerates unifo58; ye) /tkk7 wgnxqav:pedz 5rmly from 240 rpm to 360 rpm in 6.5 s. How far will a point on the edge of the wheel5e8/y;pwzkkvxn):7q 5 agedt have traveled in this time?    m

  A cooling fan is turned offnh0iroqjb6l)zm2 2c0c 3/l3i sgoqs : when it is running at 850rev/min It turns 1500 revolutions before c0 6cg/oj3 z0i2:3h lisq r)sqlobnm2it comes to a stop.
(a) What was the fan’s angular acceleration, assumed constant?    $\frac{rad}{s^2}$
(b) How long did it take the fan to come to a complete stop?    s

  The tires of a car make 65 revolutions as the car reduces itsig6vn.vvvnjghq( n; 5hi.esl+. ul )7 speed uniformly from 95km/h to 45km/h The tires have a di+(nn.iqlvh.76 lhnis.5vgg vj )euv;ameter of 0.80 m.
(a) What was the angular acceleration of the tires? $\approx$    $rad/s^2$
(b) If the car continues to decelerate at this rate, how much more time is required for it to stop?    s

  The tires of a car make 65 revolutions as the carmbb- 3f f.j8od reduces its speed uniformly from 95km/h to 45km/h The tires have 3 - o.bjbmf8fda diameter of 0.80 m.
(a) What was the angular acceleration of the tires? $\approx$    $rad/s^2$
(b) If the car continues to decelerate at this rate, how much more time is required for it to stop?    s

  A 55-kg person riding a bike puts all her wcj/gxb 990k5s j,n;fr+c lb *uq,i indeight on each pedal when climbing a hill. The pedals rotate in a circjuibgi b+ ,;cjd9 /nr*fl9q,5 xkc 0snle of radius 17 cm.
(a) What is the maximum torque she exerts?    $m \cdot N$
(b) How could she exert more torque?

  A person exerts a force of 55 N on the end of a door 74 cm wi0s 7x) 5:(zcpaegtgohy ,h) -qf n0nuode. What is the magnitude o f n uphy5a )) 7ng0o0(tq-xes:h,gcozf the torque if the force is exerted
(a) perpendicular to the door    $m \cdot N$
(b) at a 45 $^{\circ} $ angle to the face of the door?    $m \cdot N$

  Calculate the net torque about the axle of the wheel shown in F2ci c)2osup/iig. 8–39. Assume tuo2c/ 2 ipsci)hat a friction torque of 0.4 $m \cdot N$ opposes the motion.    $m \cdot N$  ----待选择----“counterclockwiseclockwise 

Two blocks, each of mass m, are attached to the ends of a massless rod which o tbq4-w+v;s6s2ewy l8d 6buw pivots as shown in Fig. 8–40. Initially the rod is held in the horizontal position and then released. Calculate the magnitude and direction t-66bd;y b +8vl2wq4weswousof the net torque on this system.

  The bolts on the cylinder head of w,uho mv( 4l3qan engine require tightening to a torque of 38 4q3, u vm(whol$m \cdot N$ If a wrench is 28 cm long, what force perpendicular to the wrench must the mechanic exert at its end?    N
If the six-sided bolt head is 15 mm in diameter, estimate the force applied near each of the six points by a socket wrench (Fig. 8–41).    N

  Determine the moment of inertia of a 10.8-kg sphere of radius 0.648 m when:w(bs i:n lp1)tp5fie the axis of rotation is through its center. p :(tfi1i) w pn5sl:be   $kg \cdot m^2$

  Calculate the moment of inertia oy f:y ,p52agut ip/z4wf*+wlmf a bicycle wheel 66.7 cm in diameter. The rim and tire have a combined mass of 1.25 kg. The mass of the hub can be ignored (why?).p4 pif*mwl w t/ :yu2+g,f5ayz    $kg \cdot m^2$

  A small 650-gram ball on the end of a th;fmgb15mbs j jv6) f66ao,j thin, light rod is rotated in a horizontal circlebmh,b6a f;fsmjo jj t65 )gv61 of radius 1.2 m. Calculate
(a) the moment of inertia of the ball about the center of the circle,    $kg \cdot m^2$
(b) the torque needed to keep the ball rotating at constant angular velocity if air resistance exerts a force of 0.020 N on the ball. Ignore the rod’s moment of inertia and air resistance.    $m \cdot N$

  A potter is shaping a botintpp:pr8 0+5i3uqw wl on a potter’s wheel rotating at constant angular speed (Fig. 8–42). The friction force between her hands at0iq+5p pnup8twr :3i nd the clay is 1.5 N total.
(a) How large is her torque on the wheel, if the diameter of the bowl is 12 cm?    $m \cdot N$
(b) How long would it take for the potter’s wheel to stop if the only torque acting on it is due to the potter’s hand? The initial angular velocity of the wheel is 1.6 rev/s, and the moment of inertia of the wheel and the bowl is 0.11 $kg \cdot m^2$.    s

  Calculate the moment of inertia of the array of point objects shown iha rxhcfqk9 e9;7gh* 6vc*zo) n Fig. 8–43 abk o h*f;ehx9c*zr)9aq6v gch7out
(a) the vertical axis,    $kg \cdot m^2$
(b) the horizontal axis. Assume m=1.8 kg,M=3.1kg and the objects are wired together by very light, rigid pieces of wire. The array is rectangular and is split through the middle by the horizontal axis.    $kg \cdot m^2$
(c) About which axis would it be harder to accelerate this array?

  An oxygen molecule consists of two oxygen atoms)+w;it .ji vyz whose total mass is $5.3 \times10^{ -26}$ kg and whose moment of inertia about an axis perpendicular to the line joining the two atoms, midway between them, is $ 1.9\times10^{-46 }$ $kg \cdot m^2$ From these data, estimate the effective distance between the atoms.    $\times10^{-10 }$ m

  To get a flat, uniform cylindrickxbooanf +z9us(6.7.jdgnpj ;4 i ;aux cm:1 qal satellite spinning at the correct rate, engineers fire four tangential rockets as shown in Fig. 8–44. If the satellite has a mass of 3600 kg and a radius of 4.z +x ( pa x9a6bo.oc ;f .;indujn7muj4s:gk1q0 m, what is the required steady force of each rocket if the satellite is to reach 32 rpm in 5.0 min? $\approx$    N(round to the nearest integer)

  A grinding wheel is a uniform cylinder with a radius onq 6by qgb/f m33k-fj fc5rsr5o496 fyf 8.50 cm and a mass of 0.580 kg. Calculate 3 q66y/mfyf4-kg3frc r 5f9bnb 5qsoj
(a) its moment of inertia about its center, $\approx$    $kg \cdot m^2$
(b) the applied torque needed to accelerate it from rest to 1500 rpm in 5.00 s if it is known to slow down from 1500 rpm to rest in 55.0 s。    $m \cdot N$

  A softball player swings a bat, acceler hr*tyw06f/7dsphve 8ating it from rest to 3 $rev/s$ in a time of 0.20 s. Approximate the bat as a 2.2-kg uniform rod of length 0.95 m, and compute the torque the player applies to one end of it.    $m \cdot N$

  A teenager pushes tangef2oqzos8zqro), ps557od 9 slntially on a small hand-driven merry-go-round and is able to accelersd ozosqo)8 75p so lfr,29q5zate it from rest to a frequency of 15 rpm in 10.0 s. Assume the merry-go-round is a uniform disk of radius 2.5 m and has a mass of 760 kg, and two children (each with a mass of 25 kg) sit opposite each other on the edge. Calculate the torque required to produce the acceleration, neglecting frictional torque. $\approx$   $m \cdot N$ What force is required at the edge?    N

  A centrifuge rotor rotating at 10,300 rpm is shut off and is eventually broa94cjpp,7lthfkafxwu1 i ab(j: ;- u,ught uniformly to rest by a frictional ta 4:p -bp x,ihjw71jfkul9tcuaa , f(;orque of 1.2 $m \cdot N$ If the mass of the rotor is 4.80 kg and it can be approximated as a solid cylinder of radius 0.0710 m, through how many revolutions will the rotor turn before coming to rest,    $rev$ how long will it take?    s

  The forearm in Fig. 8–45 acceleratesrt q-8b dp30hutl t.2v a 3.6-kg ball at 7 $m/s^2$ by means of the triceps muscle, as shown. Calculate
(a) the torque needed,    $m \cdot N$
(b) the force that must be exerted by the triceps muscle. Ignore the mass of the arm.    N

  Assume that a 1.00-kg ball is thrown solerhqo6h of2f,68nqf5-1 p,svc1 suack ly by the action of the forearm, which rf 1cr 2ohu1ao86,s6-f,pcs5 hn q kvfqotates about the elbow joint under the action of the triceps muscle, Fig. 8–45. The ball is accelerated uniformly from rest to 10 $m/s$ in 0.350 s, at which point it is released. Calculate
(a) the angular acceleration of the arm,    $rad/s^2$
(b) the force required of the triceps muscle. Assume that the forearm has a mass of 3.70 kg and rotates like a uniform rod about an axis at its end.    N

  A helicopter rotor blade can be considered a long thin rod, as shownjfv:g7nm/k p)a3h,bn in Fig. 8–46.ng hja )fnpkm 3b7/v:,
(a) If each of the three rotor helicopter blades is 3.75 m long and has a mass of 160 kg, calculate the moment of inertia of the three rotor blades about the axis of rotation.    $kg \cdot m^2$
(b) How much torque must the motor apply to bring the blades up to a speed of 5 $rev/s$ in 8.0 s?    $m \cdot N$

An Atwood’s machine consists o5xh+s1f(jy p lf two masses, $m_1$ and $m_2$ which are connected by a massless inelastic cord that passes over a pulley, Fig. 8–47. If the pulley has radius R and moment of inertia I about its axle, determine the acceleration of the masses $m_1$ and $m_2$ and compare to the situation in which the moment of inertia of the pulley is ignored. [Hint: The tensions $F_{T1}$ and $F_{T2}$ are not equal. We discussed this situation in Example 4–13, assuming for the pulley.]

  A hammer thrower accelerates the hammer fro0qtn,qmf /0f9:q lli/vb . qysm rest within four full turns (revolutions) and releases it at a sps:m, i0qtb. 0/l9lqqfyn fqv/ eed of 28 $m/s$ Assuming a uniform rate of increase in angular velocity and a horizontal circular path of radius 1.20 m, calculate
(a) the angular acceleration,    $rad/s^2$
(b) the (linear) tangential acceleration,    $m/s^2$
(c) the centripetal acceleration just before release,    $m/s^2$
(d) the net force being exerted on the hammer by the athlete just before release,    N
(e) the angle of this force with respect to the radius of the circular motion.    $^{\circ} $

  A centrifuge rotor has a moment ofbq) e,5mg 1wb2fmor p4 inertia of $3.75 \times10^{-2 }$ $kg \cdot m^2$ How much energy is required to bring it from rest to 8250 rpm?    J

  An automobile engine develops a*y1waq.0n/btvmc gt 5+,e cx t torque of 280 $m \cdot N$ at 3800 rpm. What is the power in watts and in horsepower?    W    hp

  A bowling ball of mass 7.3 kg and radius 9.0 cm rolls without slipping down zrc 4qmw:s2 ,l7hx.vxa lawxx.7:4cr,2 qlmz hvsne at 3.3 $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of the Earth with respect to2n r 6tsj5nj. iptl;lo o/p4b7 the Sun as the sum of trosln 6n25;ipj./tbl4poj7 two terms,
(a) that due to its daily rotation about its axis,$KE_{daily}$=    $\times10^{29 }$ J
(b) that due to its yearly revolution about the Sun. $KE_{yearly}$+    $\times10^{33 }$ J [Assume the Earth is a uniform sphere with $6 \times10^{ 24}$ kg and $6.4 \times10^{6 }$ m and is $1.5 \times10^{8 }$ km from the Sun.]$KE_{daily}$ + $KE_{yearly}$ =    $ \times10^{33 }$ J

  A merry-go-round has a mass of2oi ;up5ivg*kwev2z7 1640 kg and a radius of 7.50 m. How much net work is required to accelerate it from rest to a rotation rate of 1.00 e2k2g;wuv v*oipi 57z revolution per 8.00 s? Assume it is a solid cylinder.    J

  A sphere of radius 20.0 cm and mass 1.80 kg l mkx0zri( :.wr:o3czp4j (xg starts from rest and rolls without slipping: (ioz z(pc0kwr4j3gx lrx.:m down a 30.0 $^{\circ} $ incline that is 10.0 m long.
(a) Calculate its translational and rotational speeds when it reaches the bottom. $v_{CM}$ =    $\omega$ =    $rad/s$
(b) What is the ratio of translational to rotational KE at the bottom?    Avoid putting in numbers until the end so you can answer:
(c) do your answers in (a) and (b) depend on the radius of the sphere or its mass?

  Two masses, $m_1$ = 18 kg and $m_2$ = 26.5 kg are connected by a rope that hangs over a pulley (as in Fig. 8–47). The pulley is a uniform cylinder of radius 0.260 m and mass 7.50 kg. Initially, is on the ground and $m_2$ rests 3.00 m above the ground. If the system is now released, use conservation of energy to determine the speed of $m_2$ just before it strikes the ground. Assume the pulley is frictionless.    $m/s$

  A 2.30-m-long pole is balanced vertically on its tip. It starts to fall and 7no6nzg/az jvr+ d8h+evf7y ba08 op8 its lower end does not slip. What will be the speed of the upper end of tfgez 8v7on0a 87 +/adnb8 6zjprhv yo+he pole just before it hits the ground? [Hint: Use conservation of energy.]    $m/s$

  What is the angular momeei75h- 6x6gcve j h 9xovt49dsntum of a 0.210-kg ball rotating on the end of a thin string in a circle of radius 1.10 m at an angular i-6c eo4xg79 e65 hv9vhsjxtdspeed of 10.4 $rad/s$?    $kg \cdot m^2$

  (a) What is the angular momentum of a 2.8-kg uniform cylindrical grinding whe0e2s 8r c yx avfuj2+txj s62f91*eigkel of radius 18v j xfa01egx28yscer kuf +t9i2s2j*6 cm when rotating at 1500 rpm?    $kg \cdot m^2$
(b) How much torque is required to stop it in 6.0 s?    $m \cdot N$

A person stands, hands at his sidzf z)/q+xf t,de, on a platform that is rotating at a rate of 1.3rev/s If he raises his arms to a horizontal position, Fig. 8–48, the speed of ) qx+/ftdzfz, rotation decreases to 0.8 $rev/s$ (a) Why?
(b) By what factor has his moment of inertia changed?

  A diver (such as the one shown in Fig. 8–,e 5l9ey3:i y7yduzf1 mchkf929) can reduce her moment of inertia by a factor of about 3.5 when chk99chfl ,id17 zf5 yym3: eyueanging from the straight position to the tuck position. If she makes 2.0 rotations in 1.5 s when in the tuck position, what is her angular speed ($rev/s$) when in the straight position?   $rev/s$

  A figure skater can increase her spin rotation oa h6/abtdx 0:rate from an initial rate of 1.0 rev everyxh ba0 /o6:dta 2.0 s to a final rate of 3 $rev/s$ If her initial moment of inertia was 4.6 kg*$m^2$ what is her final moment of inertia? How does she physically accomplish this change?    $kg \cdot m^2$

  A potter’s wheel is rotating around a vertical axis through its2+;j+vdj;rk,b e4 vcg hz,u*. yix gpw(tw0xe center at a frequency of 1.5rev/s The wheel can be considered a uniform disk of mass 5.0 kg and diameter 0.40 m. The potter then throws a 3.1-kg chunk of clay, approximately shaped as a flat z+ti0pgcj;.b4hjxke+ v ,dr2v;,uyg w wex* (disk of radius 8.0 cm, onto the center of the rotating wheel. What is the frequency of the wheel after the clay sticks to it?    $rev/s$

  (a) What is the angulugm ii4a j np5 af53(-hbbf:s,ar momentum of a figure skater spinning at 3.5 $rev/s$ with arms in close to her body, assuming her to be a uniform cylinder with a height of 1.5 m, a radius of 15 cm, and a mass of 55 kg?    $kg \cdot m^2$
(b) How much torque is required to slow her to a stop in 5.0 s, assuming she does not move her arms?    $m \cdot N$

  Determine the angular mom5. kkfeub *c hoxfc(::x5 ozfu5vu+9s /gz)fg entum of the Earth
(a) about its rotation axis (assume the Earth is a uniform sphere),    $\times 10^{33} \; kg \cdot m^2$

(b) in its orbit around the Sun (treat the Earth as a particle orbiting the Sun). The Earth has mass $6 \times 10^{24} \; kg$ and radius $6.4 \times 10^{6} \; m$ and is $1.5 \times 10^{8} \; km$ from the Sun.    $\times10^{40} \; kg \cdot m^2$

A nonrotating cylindricalq c53/rnenp*n96cj3pweyz.5wwdjv c/8 g3 xq disk of moment of inertia I is dropped onto an identical disk ro6/y* qpqd3 c8p3w/rzgc5nj9.j5we evcxnw3n tating at angular speed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns(.;b2 f*dzbb 3evpdzh i5t8g r at 2.4 $rev/s$ around a frictionless spindle. A nonrotating rod, of the same mass as the disk and length equal to the disk’s diameter, is dropped onto the freely spinning disk, Fig. 8–49. They then both turn around the spindle with their centers superposed. What is the angular frequency in rev/s of the combination?    $rev/s$

  A person of mass 75 kg stands at the center of a rotating o4mohv0b9 xo6merry-go-round platform of radius 3.0 m and moment of inertb 9o604mo xvhoia 920 $kg \cdot m^2$ The platform rotates without friction with angular velocity 2 $rad/s$ The person walks radially to the edge of the platform.
(a) Calculate the angular velocity when the person reaches the edge.    $rad/s$
(b) Calculate the rotational kinetic energy of the system of platform plus person before and after the person’s walk.$KE_i$ =    J $KE_f$ =    J

  A 4.2-m-diameter merry-go-rdfjif6tf)a3 np45 (buound is rotating freely with an angular velocity of 0.( 45 3 jb)atfndip6fuf8 $rad/s$ Its total moment of inertia is 1760 $kg \cdot m^2$ Four people standing on the ground, each of mass 65 kg, suddenly step onto the edge of the merry-go-round. What is the angular velocity of the merry-go-round now?    $rad/s$ What if the people were on it initially and then jumped off in a radial direction (relative to the merry-go-round)?    $rad/s$

  Suppose our Sun eventually collapses into;(o/b )w/felnnkdp+narjfp : j+fz5 5 a white dwarf, losing about half its mass in the process, and winding up with a radius 1.0% of its existing radius. Assuming the lost mass carries away no angular momentum, what would the Sun’s new rotation rat/pz;fpjf / 5nr a5) w+ol be+nfjk(:dne be?(round to the nearest integer)$\approx$    $rad/s$ (Take the Sun’s current period to be about 30 days.) What would be its final KE in terms of its initial KE of today?$KE_{f}$=    $KE_{i}$

  Hurricanes can involve x, fc1rexi o1hu6zct: wr2-;enk(: sj winds in excess of 120 $km/h$ at the outer edge. Make a crude estimate of
(a) the energy,    $ \times10^{16 }$ J
(b) the angular momentum, of such a hurricane, approximating it as a rigidly rotating uniform cylinder of air (density 1.3 $kg \cdot m^2$) of radius 100 km and height 4.0 km.    $ \times10^{20 }$ $kg \cdot m^2$

  An asteroid of mass qgd3(, wr1cbr$ 1.0\times10^{ 5}$ traveling at a speed of relative to the Earth, hits the Earth at the equator tangentially, and in the direction of Earth’s rotation. Use angular momentum to estimate the percent change in the angular speed of the Earth as a result of the collision.    $\times10^{-16 }$ %

A person stands on a platform, initially at rest, that can rotate freely witho (g mh rqeb6r h cb9)e1hi1x*;jb)9lmxut friction. The momentq*j;r r6l (chbxe9eh)m1g) 91ibxm bh of inertia of the person plus the platform is $I_P$ The person holds a spinning bicycle wheel with its axis horizontal. The wheel has moment of inertia $I_W$ and angular velocity $\omega_W$ What will be the angular velocity $\omega_W$ of the platform if the person moves the axis of the wheel so that it points (a) vertically upward, (b) at a 60º angle to the vertical, (c) vertically downward? (d) What will $\omega_P$ be if the person reaches up and stops the wheel in part (a)?

  Suppose a 55-kg person stands at the edge of a 6.5-mb kcj018 nrve4 diameter merry-go-round turntable that is mounted81nkjc4 ev br0 on frictionless bearings and has a moment of inertia of 1700 $kg \cdot m^2$ The turntable is at rest initially, but when the person begins running at a speed of 3.8 $m/s$ (with respect to the turntable) around its edge, the turntable begins to rotate in the opposite direction. Calculate the angular velocity of the turntable.    $rad/s$

A large spool of rope rolls on the ground with the end of tu a4;z+for rwgt1;970m0znx4v z yr bmhe rope lying on the top edge of the spool. A person grabs the end of the rope an0;xg mvaou m1 yt4+fwb709rzn 4z rzr;d walks a distance L, holding onto it, Fig. 8–50. The spool rolls behind the person without slipping. What length of rope unwinds from the spool? How far does the spool’s center of mass move?

  The Moon orbits the Earth such that the same side always faces the Eart+ b9tquaz-5p9 x l9+ pbjsus0mh. Determine the ratio of the Moon’s spin angular momentum (about its own axis) to its orbital angular momentum. (pq z 0p+j9xs 9al 95t+-umbbsuIn the latter case, treat the Moon as a particle orbiting the Earth.)    $\times10^{ -6}$

  A cyclist accelerates from rest at a rate of 1 mf) ,oy0n-:z nzvo gs -ed3xq7o0mbi .e/$s^2$ How fast will a point on the rim of the tire at the top be moving after 3.0 s? [Hint: At any moment, the lowest point on the tire is in contact with the ground and is at rest — see Fig. 8–51.]    $m/s$

  A 1.4-kg grindstone in the shape of a uniform cylinder of radius k 4y we:,me;qt. 86b-z8olull/ml afi0.20 m acquires a rotational ra/tlk-y za6i m :feuw,;l8.e4bl lmoq8te of from rest over a 6.0-s interval at constant angular acceleration. Calculate the torque delivered by the motor.    $m \cdot N$

  (a) A yo-yo is made of two solid cylindrical disks, each of mass 0.050 kg k3ycgzmms4s f : rq7fc-46ie2+x+dt:0zt n mband diameter 0.075 m, joinekn4srcixmsbct3:q-m70 yz4tzfgm +f2d: e+ 6d by a (concentric) thin solid cylindrical hub of mass 0.0050 kg and diameter 0.010 m. Use conservation of energy to calculate the linear speed of the yo-yo when it reaches the end of its 1.0-m-long string, if it is released from rest.    $m/s$
(b) What fraction of its kinetic energy is rotational?    %

  (a) For a bicycle, how is the angular speed of the rear wheep3 p6 nixxsb 8j+dp;cme2n- :il ($\omega_R$) related to that of the pedals and front sprocket ($\omega_F$) Fig. 8–52? That is, derive a formula for ($\omega_R$)/($\omega_F$) Let $N_F$ and $N_R$ be the number of teeth on the front and rear sprockets, respectively. The teeth are spaced equally on all sprockets so that the chain meshes properly.
(b) Evaluate the ratio ($\omega_R$)/($\omega_F$) when the front and rear sprockets have 52 and 13 teeth, respectively,   
(c) when they have 42 and 28 teeth.   

  Suppose a star the size of our Sun, but with mass 8.0 tim15m6 1- ,kxpyg bks aj)ila.wtes as great, were rotating at a speed of 1.0 revolution every 12 days. If it were to undergo gravitational collapse to a neutron star of radius 11 km, losing three-quarters of its mass in the process, what would its rotation speed be? Assumem-)b.1ayjil5tk6wkp agxs,1 that the star is a uniform sphere at all times, and that the lost mass carries off no angular momentum.    $\times10^{9 }$ $rev/day$

  One possibility for a low-aw0i.8 k /6k d 50ev7ioewcolopollution automobile is for it to use energy stored in a heavy rotating flywheel. Suppose such a car has a total mass of 1400 kg, uses a uniform cylindrical flywheolkovd5eo. i0 6 w8wck/a 07ieel of diameter 1.50 m and mass 240 kg, and should be able to travel 350 km without needing a flywheel “spinup.”
(a) Make reasonable assumptions (average frictional retarding force = 450N twenty acceleration periods from rest to equal uphill and downhill, and that energy can be put back into the flywheel as the car goes downhill), and show that the total energy needed to be stored in the flywheel is about $ 1.7\times10^{8 }$J.    $ \times10^{ 8}$ J
(b) What is the angular velocity of the flywheel when it has a full “energy charge”?    $rad/s$
(c) About how long would it take a 150-hp motor to give the flywheel a full energy charge before a trip? $\approx$    min

  Figure 8–53 illustrates vxo7mu*r3b;l +3xo pman $H_2O$ molecule. The O–H bond length is 0.96 nm and the H–O–H bonds make an angle of 104 $^{\circ} $. Calculate the moment of inertia for the $H_2O$ molecule about an axis passing through the center of the oxygen atom
(a) perpendicular to the plane of the molecule,    $\times10^{-45 }$ $kg \cdot m^2$
(b) in the plane of the molecule, bisecting the H–O–H bonds.    $ \times10^{-45 }$ $kg \cdot m^2$

  A hollow cylinder (hoop) is rolling on a horizontal surface 0,jy gfs2osxtrl ,s 54w3 qhe0at speed v=3.3 $m/s$ when it reaches a 15 $^{\circ} $ incline.
(a) How far up the incline will it go? $\approx$    m (round to one decimal place)
(b) How long will it be on the incline before it arrives back at the bottom?    s

A uniform rod of mass M and length L can pivot freely (i.e., w;z1ji:v h5h5cjnw gn) :z.t mwe ignore friction)ctw 5;j1zz.n :gw:miv5hj h)n about a hinge attached to a wall, as in Fig. 8–54. The rod is held horizontally and then released. At the moment of release, determine (a) the angular acceleration of the rod, and (b) the linear acceleration of the tip of the rod. Assume that the force of gravity acts at the center of mass of the rod, as shown. [Hint: See Fig. 8–21g.]

A wheel of mass M has radius 7/dpxwxmc0 * mR. It is standing vertically on the floor, and we want to exert a horizontal force F at its axle so that it will climb a stemcpmw xx0* d/7p against which it rests (Fig. 8–55). The step has height h, where h

  A bicyclist traveling with speed v=4gwe:l2q(tkg 7.2m/s on a flat road is making a turn with a radius The forces acting on the cyclist and cycle are the normal fkeg(l:q72wgt orce $\left(\mathbf{\vec{F}}_{\mathrm{N}}\right)$ and friction force $\left(\mathbf{\vec{F}}_{\mathbf{fr}}\right)$ exerted by the road on the tires, and $m\vec{\mathbf{g}}$ the total weight of the cyclist and cycle (see Fig. 8–56).
(a) Explain carefully why the angle $\theta$ the bicycle makes with the vertical (Fig. 8–56) must be given by tan $\tan\theta=F_{\mathrm{fr}}/F_{\mathrm{N}}$ if the cyclist is to maintain balance.(round to the nearest integer)
(b) Calculate $\theta$ for the values given.    $^{\circ} $
(c) If the coefficient of static friction between tires and road is $\mu_s=0.70$ what is the minimum turning radius?    m

  Suppose David puts a 0.50-kg rock into a sling of length 1.5 m and begi: ykj nudv0+8s 9hb1zg 80qzkyns whirling the rock in a nearly horizontal circle above his head, accelerating it from rest to a rate of 120 rpm after 5.0 s. What is the torque required to achieve this feat, and where does the torque0j: v 9k8ygu0s+n8hzky q1zdb come from?    $m \cdot N$

  Model a figure skater’s bodytr)1 t;xrl vh*7jm-sv as a solid cylinder and her arms as thin rods, making reasonable estimates for the dimensions. Then 1 t7mv v*r l)hjx;r-tscalculate the ratio of the angular speeds for a spinning skater with outstretched arms, and with arms held tightly against her body.   

  You are designing a clutch assembly whi ,pqyfvg.ba,-fbxi c03q 3j:kch consists of two cylindrical plates, of p.cxk0 ,,q:fvg3qabf-3i ybjmass $M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ and $M_{\mathrm{B}}=9.0$ $\mathrm{kg}$ with equal radii R=0.60 $\mathrm{m}$ They are initially separated (Fig. 8–57). Plate $M_{\mathrm{A}}$ is accelerated from rest to an angular velocity $\omega_1=7.2$ $\mathrm{rad/s}$ in time $\Delta t=2.0$ s Calculate
(a) the angular momentum of $M_{\mathrm{A}}$    $kg \cdot m^2$
(b) the torque required to have accelerated $M_{\mathrm{A}}$ from rest to $\omega_{1}$    $m \cdot N$
(c) Plate $M_{\mathrm{B}}$ initially at rest but free to rotate without friction, is allowed to fall vertically (or pushed by a spring), so it is in firm contact with plate $M_{\mathrm{A}}$ (their contact surfaces are high-friction). Before contact, $M_{\mathrm{A}}$ was rotating at constant $\omega_{1}$ After contact, at what constant angular velocity $\omega_{s}$ do the two plates rotate?    $rad/s$

  A marble of mass m and radius r rolls along the looped roug 0zjwn*/ qbk85cap )60,himg 7nspfc bh track of Fig. 8–58. What is the minimum value of the vertical height h that the marble g70,zfc5in m6cp0p8)/hnbsa w kjbq*must drop if it is to reach the highest point of the loop without leaving the track? Assume $r\ll R$ and ignore frictional losses. h =    R

  Repeat Problem 84, bu f f jsa(5 my3x2(ah5ni*evkihq63 1rlt do not assume $r\ll R$ h =    (R-r)

  The tires of a car make 85 revolutions as jza((z jtbm*b n.a 9e-the car reduces its speed uniformly from 90km/h to 60km/h The tires have a diameter of 0.90 m. (a) What was the anbjaejta-zn( .m (9*zb gular acceleration of each tire? $\approx$    $rad/s^2$(round to two decimal place)
(b) If the car continues to decelerate at this rate, how much more time is required for it to stop?    s